Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Where are nature’s missing structures?


Our society’s environmental and economic progress depends on the development of high-performance materials such as lightweight alloys, high-energy-density battery materials, recyclable motor vehicle and building components, and energy-efficient lighting. Meeting these needs requires us to understand the central role of crystal structure in a material’s properties. Despite more than 50 years of progress in first-principles calculations, it is still impossible in most materials to infer ground-state properties purely from a knowledge of their atomic components—a situation described as ‘scandalous’ in the well-known essay by Maddox1. Many methods attempt to predict crystal structures and compound stability, but here I take a different tack—to infer the existence of structures on the basis of combinatorics and geometric simplicity2. The method identifies ‘least random’ structures, for which the energy is an extremum (maximum or minimum). Although the key to the generic nature of the approach is energy minimization, the extrema are found in a chemistry-independent way.

Your institute does not have access to this article

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: All 17 geometrically possible structures with four or fewer atoms per unit cell.
Figure 2: The average bond type for the L10 structure.
Figure 3: Deviations of bond averages.
Figure 4: Likelihood measures (total bond averages) for each compound, ranked in descending order.
Figure 5: Likelihood measures for b.c.c.-based structures with four atoms per cell or fewer.


  1. Maddox, J. Crystals from first principles. Nature 335, 201 (1988).

    Article  Google Scholar 

  2. Chaikin, P. Random thoughts. Phys. Today 8–9 (June 2007).

    Article  Google Scholar 

  3. Hume-Rothery, W. Researches on the nature, properties, and condition of formation of intermetallic compounds. J. Inst. Met. 35, 319–325 (1926).

    Google Scholar 

  4. Hume-Rothery, W. Phase Stability in Metals and Alloys 3–23 (McGraw Hill, New York, 1967).

    Google Scholar 

  5. Miedema, A. R., de Boer, F. R. & de Chatel, P. F. Empirical description of the role of electronegativity in alloy formation. J. Phys. F 3, 1558–1576 (1973).

    CAS  Article  Google Scholar 

  6. Pettifor, D. G. The structures of binary compounds. I. Phenomenological structure maps. J. Phys. C 19, 285–313 (1986).

    CAS  Article  Google Scholar 

  7. Curtarolo, S., Morgan, D. & Ceder, G. Accuracy of ab initio methods in predicting the crystal structures of metals: review of 80 binary alloys. Calphad 29, 163 (2005).

    CAS  Article  Google Scholar 

  8. Fischer, C. C., Tibbetts, K. J., Morgan, D. & Ceder, G. Predicting crystal structures by merging data mining with quantum mechanics. Nature Mater. 5, 641–646 (2006).

    CAS  Article  Google Scholar 

  9. Sanchez, J. M., Ducastelle, F. & Gratias, D. Generalized cluster description of multicomponent systems. Physica A 128, 334–350 (1984).

    Article  Google Scholar 

  10. de Fontaine, D. Cluster approach to order-disorder transformations in alloys. Solid State Phys. 47, 33–176 (1994).

    Article  Google Scholar 

  11. Zunger, A. in Statics and Dynamics of Alloy Phase Transitions (eds Turchi, P. E. A. & Gonis, A.) 361–419 (NATO ASI Series, Ser. B, Plenum, New York, 1994).

    Book  Google Scholar 

  12. Soven, P. Coherent-potential model of substitutional disordered alloys. Phys. Rev. 156, 809–813 (1967).

    CAS  Article  Google Scholar 

  13. Gonis, A. et al. Configurational energies and effective cluster interactions in substitutionally disordered binary alloys. Phys. Rev. B 36, 4630–4646 (1987).

    CAS  Article  Google Scholar 

  14. Pettifor, D. G. New many-body potential for the bond order. Phys. Rev. Lett. 63, 2480–2483 (1989).

    CAS  Article  Google Scholar 

  15. Daw, M. S. & Baskes, M. I. Semiempirical, quantum mechanical calculation of hydrogen embrittlement in metals. Phys. Rev. Lett. 50, 1285–1288 (1983).

    CAS  Article  Google Scholar 

  16. Daw, M. S. & Baskes, M. I. Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B 29, 6443–6453 (1984).

    CAS  Article  Google Scholar 

  17. Sluiter, M. H. F. Some observed bcc, fcc, and hcp superstructures. Phase Transit. 80, 299–309 (2007).

    CAS  Article  Google Scholar 

  18. Santoro, A. & Mighell, A. D. Properties of crystal lattices: The derivative lattices and their determination. Acta Crystallogr. A 28, 284–287 (1972).

    Article  Google Scholar 

  19. Santoro, A. & Mighell, A. D. Coincidence-site lattices. Acta Crystallogr. A 29, 169–175 (1973).

    CAS  Article  Google Scholar 

  20. Hart, G. L. W. A new intermetallic prototype? Verifying new structure predictions in CdPt and PtPd, in Bulletin of the March Meeting (American Physical Society, 2007,

  21. Pietrokowsky, P. Novel ordered phase, Pt8Ti. Nature 206, 291 (1965).

    CAS  Article  Google Scholar 

  22. Ducastelle, F. Order and Phase Stability in Alloys (Cohesion and Structure, Vol. 3, North-Holland, Amsterdam, 1991).

    Google Scholar 

  23. Garbulsky, G. D., Tepesch, P. D. & Ceder, G. in Materials Theory and Modeling (eds Broughton, J., Bristowe, P. & Newsam, J.) 259–265 (Materials Research Society, Pittsburg, 1993).

    Google Scholar 

  24. Kanamori, J. & Kakehashi, Y. Conditions for the existence of ordered structure in binary alloy systems. J. Physique 38, 274 (1977).

    Google Scholar 

  25. Allen, S. M. & Cahn, J. W. Ground state structures in ordered binary alloys with second neighbor interactions. Acta Metallurg. 20, 423–433 (1972).

    CAS  Article  Google Scholar 

  26. Sanchez, J. M. & de Fontaine, D. in Structure and Bonding in Crystals Vol. II (eds O’Keeffe, M. & Navrotsky, A.) 117–132 (Academic, New York, 1981).

    Book  Google Scholar 

  27. Drautz, R. Ab-initio Statistical Mechanics for Ordering and Segregation at the (110) Surface of Ni90%-Al. Thesis, Max-Plank-Institut für Metallforshung (2003).

Download references


The author gratefully acknowledges financial support from the National Science Foundation through Grant Nos. DMR-0244183 and DMR-0650406. I would like to thank the following individuals for fruitful discussions: R. Drautz, S. Bärthlein, S. Müller, M. Leone and B. Kolb. Input from V. Blum was particularly useful.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Gus L. W. Hart.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hart, G. Where are nature’s missing structures?. Nature Mater 6, 941–945 (2007).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing